Endomorphisms of abelian groups and the theorem of Baer and Kaplansky

May, Warren; Toubassi, Elias

doi:10.1016/0021-8693(76)90139-3

Cited by 38 publications

(9 citation statements)

References 2 publications

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“…55] and, as shown in [35,Theorem 3.3], and in mixed groups the endomorphism rings E(A) and E(tA) are isomorphic if and only if A is a fully invariant subgroup of the cotorsion hull (tA) • .…”

Section: Full Transitivity Of Algebraically Compact Groupsmentioning

confidence: 96%

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The Lattice of Fully Invariant Subgroups of a Cotorsion Group

Kemoklidze

2014

J Math Sci

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Section: Full Transitivity Of Algebraically Compact Groupsmentioning

confidence: 96%

“…112]). In [35] it is shown that in a reduced Abelian group A with torsion part T any endomorphism of the group T uniquely extends to the endomorphism A if and only if A is a pure fully invariant subgroup of the cotorsion hull T • = Ext(Q/Z, T ) of the group T .…”

Section: On Pure Fully Invariant Subgroups Of a Cotorsion Hullmentioning

confidence: 99%

The Lattice of Fully Invariant Subgroups of a Cotorsion Group

Kemoklidze

2014

J Math Sci

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“…It is noteworthy that endomorpohisms in cotorsion groups are completely defined by their action on the torsion part and, as shown by W. May and E. Toubassi [3], for a mixed group A the ring of endomorphisms ( ) E A is isomorphic to ( ) E tA if and only if A is a fully invariant subgroup of the cotorsion hull ( )…”

Section: ( )mentioning

confidence: 98%

On the Full Transitivity of a Cotorsion Hull of the Pierce Group

Kemoklidze¹

2014

APM

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The paper considers the problem of full transitivity of a cotorsion hull G • of a separable primary group G when a ring of endomorphisms () E G of the group G has the form () s p E G J ⊕ , where () s E G is a subring of small endomorphisms of the ring () E G , whereas p J is a ring of integer p-adic numbers. Investigation of the issue of full transitivity of a group is essentially helpful in studying its fully invariant subgroups as well as the lattice formed by these subgroups. It is proved that in the considered case, the cotorsion hull is not fully transitive. A lemma is proposed, which can be used in the study of full transitivity of a group and in other cases.

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“…p Though the notion of a cotorsion group and its generalizations are studied sufficiently well (see [15][16][17][18]), little is known about the lattice of fully invariant subgroups of a cotorsion group. The investigation of this problem in the case of a cotorsion hull is important because endomorphisms in this class of groups are completely defined by their action on the torsion part and, as shown in [19], for mixed groups the ring of endomorphisms is isomorphic to the ring of endomorphisms of the torsion part if and only if the group is a fully invariant subgroup of the cotorsion hull of its torsion part. The study of the lattice of fully invariant subgroups makes essential use of the notions of an indicator and a fully transitive group.…”

Section: P Pmentioning

confidence: 99%

The Lattice of Fully Invariant Subgroups of the Cotorsion Hull

Kemoklidze¹

2013

APM

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The paper considers the lattice of fully invariant subgroups of the cotorsion hull T  when a separable primary group is an arbitrary direct sum of torsion-complete groups.The investigation of this problem in the case of a cotorsion hull is important because endomorphisms in this class of groups are completely defined by their action on the torsion part and for mixed groups the ring of endomorphisms is isomorphic to the ring of endomorphisms of the torsion part if and only if the group is a fully invariant subgroup of the cotorsion hull of its torsion part. In the considered case, the cotorsion hull is not fully transitive and hence it is necessary to introduce a new function which differs from an indicator and assigns an infinite matrix to each element of the cotorsion hull. The relation difined on the set T   of these matrices is different from the relation proposed by the autor in the countable case and better discribes the lower semilattice . The use of the relation essentially simplifies the verification of the required properties. It is proved that the lattice of fully invariant subgroups of the group T   is isomorphic to the lattice of filters of the lower semilattice  .

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Endomorphisms of abelian groups and the theorem of Baer and Kaplansky

Cited by 38 publications

References 2 publications

The Lattice of Fully Invariant Subgroups of a Cotorsion Group

The Lattice of Fully Invariant Subgroups of a Cotorsion Group

On the Full Transitivity of a Cotorsion Hull of the Pierce Group

The Lattice of Fully Invariant Subgroups of the Cotorsion Hull

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